A key strategy employed in the DFGK model [1]} is the introduction of a superpotential-inspired function \(W[\hat{\phi }]\)  which is used to simplify the stabilized model's background field equations (Eqs. (REF ), (REF ), (REF ), and (REF )). In particular, it is assumed that the scalar bulk and brane potentials may be parameterized as:
\(&\hspace{-60.0pt}V r_{c}^{2} = \dfrac{1}{8}\bigg (\dfrac{dW}{d\hat{\phi }}\bigg )^{2} - \dfrac{W^2}{24}~,\\V_{1}r_{c} = +\dfrac{W}{2} + \beta _{1}^{2}\bigg [\hat{\phi }(\varphi ) -\phi _{1}\bigg ]^2~,\hspace{20.0pt}&\hspace{20.0pt}V_{2}r_{c} = -\dfrac{W}{2} + \beta _{2}^{2}\bigg [\hat{\phi }(\varphi )-\phi _{2}\bigg ]^2~, \) 
